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Population based self-renewal model

I want a model that shares one $b(\text{NSC})$ for both populations, and I want to see if that fits.

Plot of b against NSC

For workability I will use a $b(\text{NSC})$ instead of the other canditates. This will make it much easier to estimate initial values (as $b$ is dependent on $\text{NSC}_0$ and not the ratio which we need to estimate)

What to choose as a shape for $b$?

Negative log10 has some good support, as this seems to be the relationship in these two fits:

Log linear

This does however have the problem of being technically able to go $ b>0.5 $, which would. In reality I do think that it will however stop before then, as the stem cell population has sufficiently decreased.

Unfortunately this log-linear relationship can produce negative values of $b$ given sufficient $\text{NSC}$. As $b$ is a probability, this is not really meaningful and thus this log-linear relationship is only really viable as an approximation with sufficiently small $\text{NSC}$ or a suficient set of parameters.

Sigmoid

A sigmoid is another viable alternative for this relationship. In the simulations we have thus far we have only ever explored one half of this sigmoid and thus there isn't really too much evidence for this in fact being sigmoidally shaped. A sigmoid is however supported by the argument that $b<0$ is meaningless and thus we need a function that (similar to the previous function) $$ [0,\infty) \rightarrow [0, 0.5] $$ or something close $$ [0,\infty) \rightarrow [0, 0.5) $$

The most convenient example of this is probably the Hill function (as this has use in biochemical modelling and is frequently used with some concentration as a support and a fraction as an output, in essence the same type of function we are looking for here).

It appears to be expressed as: $$ y = \frac{x^n}{K_d+x^n} $$ With ligand concentrations $x$, fraction bound $y$, dissociation constant $K_d$ and Hill coefficient $n$. This gives us two parameters to determine the shape: $K_d$ determines the general location of the function and $n$ determines the 'steepness' of it.

This second parameterisation is nicer as $K_a$ represents the half-occupancy concentration, or for us the number of stem cells needed for $b=0.5$, i.e. the steady state or the number of stem cells we will plateau at.

There is two possible problems with this:

  1. The function takes values $ >0.5 $. For the dynamics this should not be a problem as at that point the population will be in a steady-state anyways. It may however be a problem as the shape cannot match that of the old $b$ function.
  2. The function is backwards. Self-renewal probability should be going down as more stem cells are present.

We can of course easily fix the second issue:

And eye-balling a fit, it seems to be plausible that we can fit this function:

Although I would really like this to be parameterised with a slope instead, this will do for now.

Model definition

WT

Model fit

Model plot

IFNKO

Model fit

Model plot

Combined b

Fit with other genotypes self-renewal

KO

Model plot

WT

Model plot

Interactive

Cofit

Combined parameters

Separate parameters

Partially separate

Plot

Partial Cofit

Conclusion

Fitting one model with the other model's $b(\text{NSC})$ doesn't produce good fits. Fitting both models at once with a shared $b(\text{NSC})$ also doesn't produce good fits. Thus the hypothesis that there must be a shared $b(\text{NSC})$ is probably false (I would alse really like to have something interactive for this, but Interact.jl is broken). There might still be a bunch of other things that can be true:

The second I could investigate by bayesian inference or some form of crossvalidation.

Turing PPL inferrences